[论文解读] Network Cournot Competition

Melika Abolhassani, MohammadHossein Bateni+3 more|arXiv (Cornell University)|May 8, 2014

Game Theory and Applications参考文献 13被引用 7

一句话总结

本文提出了网络古诺竞争（Network Cournot Competition, NCC），一种基于双分图接入网络的博弈论模型，其中企业通过网络在多个市场间竞争。当成本函数为凸函数、价格函数为线性或强单调函数时，该文提出了一种组合算法，可在多项式时间内计算出纯纳什均衡，实现了超越以往基于优化方法的唯一性与效率。

ABSTRACT

Cournot competition is a fundamental economic model that represents firms competing in a single market of a homogeneous good. Each firm tries to maximize its utility---a function of the production cost as well as market price of the product---by deciding on the amount of production. In today's dynamic and diverse economy, many firms often compete in more than one market simultaneously, i.e., each market might be shared among a subset of these firms. In this situation, a bipartite graph models the access restriction where firms are on one side, markets are on the other side, and edges demonstrate whether a firm has access to a market or not. We call this game \emph{Network Cournot Competition} (NCC). In this paper, we propose algorithms for finding pure Nash equilibria of NCC games in different situations. First, we carefully design a potential function for NCC, when the price functions for markets are linear functions of the production in that market. However, for nonlinear price functions, this approach is not feasible. We model the problem as a nonlinear complementarity problem in this case, and design a polynomial-time algorithm that finds an equilibrium of the game for strongly convex cost functions and strongly monotone revenue functions. We also explore the class of price functions that ensures strong monotonicity of the revenue function, and show it consists of a broad class of functions. Moreover, we discuss the uniqueness of equilibria in both of these cases which means our algorithms find the unique equilibria of the games. Last but not least, when the cost of production in one market is independent from the cost of production in other markets for all firms, the problem can be separated into several independent classical \emph{Cournot Oligopoly} problems. We give the first combinatorial algorithm for this widely studied problem.

研究动机与目标

• 建模企业通过网络结构仅接入部分市场的多市场竞争。
• 为此类网络古诺博弈开发高效的纯纳什均衡计算算法。
• 建立均衡唯一且可在多项式时间内计算的条件。
• 为经典古诺寡头垄断提供一种组合方法，替代以往基于优化的方法。

提出的方法

• 当价格函数为线性时，为NCC设计一个势函数，使均衡计算可转化为单函数优化，而非多个企业效用的联合优化。
• 将非线性情形建模为非线性互补问题（NCP），以利用强凸性与单调性。
• 通过总产量的二分查找，高效定位均衡总产出。
• 应用Topkis单调性定理，证明最优响应集的单调性，从而验证二分查找逻辑的正确性。
• 利用企业收益函数的超模性，确保搜索过程的正确性。
• 设计一种组合算法，时间复杂度为O(n log²(Q_max))，显著快于以往方法。

实验结果

研究问题

• RQ1 当价格函数为线性时，能否为网络古诺竞争构建势函数，从而实现高效均衡计算？
• RQ2 非线性价格与成本函数需满足何种条件，才能保证NCC中纯纳什均衡的存在性与唯一性？
• RQ3 在联网古诺设定下，如何高效计算均衡总产量？
• RQ4 组合算法能否在经典古诺寡头垄断中超越现有基于优化的方法？
• RQ5 哪类价格函数能确保收益函数的强单调性，从而实现多项式时间求解？

主要发现

• 当价格函数为线性时，NCC存在势函数，可通过单函数优化实现均衡计算。
• 对于强凸成本函数与强单调收益函数，多项式时间算法可计算出唯一均衡。
• 该算法时间复杂度为O(n log²(Q_max))，其中Q_max为最大可能总产量。
• 确保强单调性的价格函数类别包含广泛函数族，超越线性函数。
• 所提出的组合算法相较于经典古诺寡头垄断的以往优化方法，更为简洁且高效。
• 在所研究条件下，均衡具有唯一性，且该算法能正确识别唯一均衡。